Characterizations of weighted core inverse in rings with involution

$R$ is a unital ring with involution. We investigate the characterizations and representations of weighted core inverse an element in by idempotents units. For example, let $a\in R$ $e\in be invertible Hermitian element, $n\geqslant 1$, then $a$ $e$-core if only there exists (or idempotent) $p$ such that $(ep)^{\ast}=ep$, $pa=0$ $a^{n}+p$ $a^{n}(1-p)+p$) invertible. As consequence, $e, f\in two elements, weighted-$\mathrm{EP}$ respect to $(e, f)$ $(fp)^{\ast}=fp$, $pa=ap=0$ These results generalize improve conclusions \cite{Li}.